L(s) = 1 | − 2-s − 3-s + 4-s − 2.06·5-s + 6-s − 4.35·7-s − 8-s + 9-s + 2.06·10-s − 11-s − 12-s − 0.682·13-s + 4.35·14-s + 2.06·15-s + 16-s − 1.26·17-s − 18-s + 0.110·19-s − 2.06·20-s + 4.35·21-s + 22-s + 4.31·23-s + 24-s − 0.716·25-s + 0.682·26-s − 27-s − 4.35·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.925·5-s + 0.408·6-s − 1.64·7-s − 0.353·8-s + 0.333·9-s + 0.654·10-s − 0.301·11-s − 0.288·12-s − 0.189·13-s + 1.16·14-s + 0.534·15-s + 0.250·16-s − 0.307·17-s − 0.235·18-s + 0.0253·19-s − 0.462·20-s + 0.949·21-s + 0.213·22-s + 0.900·23-s + 0.204·24-s − 0.143·25-s + 0.133·26-s − 0.192·27-s − 0.822·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08518184276\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08518184276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 + 2.06T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 13 | \( 1 + 0.682T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 0.110T + 19T^{2} \) |
| 23 | \( 1 - 4.31T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 + 7.84T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 7.98T + 53T^{2} \) |
| 59 | \( 1 + 7.49T + 59T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 9.82T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 1.79T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 1.93T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.494238138843761731455157261456, −7.57687491222293543751218453128, −7.03243670009285074013776552612, −6.46804289077951762999349910870, −5.63409516189929961669365721566, −4.72372828494592495272281110653, −3.54262276020231468697099951752, −3.16529647826868001574003099819, −1.76129446741256723512720010298, −0.18295121727182309360911537422,
0.18295121727182309360911537422, 1.76129446741256723512720010298, 3.16529647826868001574003099819, 3.54262276020231468697099951752, 4.72372828494592495272281110653, 5.63409516189929961669365721566, 6.46804289077951762999349910870, 7.03243670009285074013776552612, 7.57687491222293543751218453128, 8.494238138843761731455157261456